Section 10.1: Sequences
Def: A sequence is an ordered list of numbers a1 , a2 , a3 ,...,an , .... The
sequence {a1 , a2 , a3 ,..,a100 , ..} is often denoted by {an }∞
n=1 or {an }. The domain
of a sequence is a subset of all non negative integers, usually indexed by n.
We will deal with infinite sequences and so each term an will have a successor,
an+1 .
1. Write out the first 5 terms of the sequence
n+1
n+3
∞
.
n=1
2. Find a general formula for the sequence:
1 2
3 4
a.) − , , − ,
,...
4 9
16 25
b.) 1,
1 1 1
1
, ,
,
,...
2 6 24 120
c.) A set of consecutive even integers.
d.) A set of consecutive odd integers, whose terms alternate signs.
Def: If n→∞
lim an = L, then we say the sequence {an } converges to L. If
lim an = ∞ or does not exist, then we say the sequence {an } diverges.
n→∞
Note: If a sequence diverges, it is necessary to describe the divergence.
3. Determine whether the following sequences converge or diverge. If it converges,
find the limit. If it diverges, explain why.
a.) an = n
b.) an = (−1)n
c.) an =
s
4n + 3
7n + 6
d.) an = arccos
e.) an =
−n + 1
2n + 3
1
1
ln(3n + 1) − ln(4n)
3
3
f.) an =
n
ln n
g.) an =
cos n
n3
h.) an =
n
n!
i.)
3 4 5 6
, , , , ...
16 25 36 49
j.) {0, 2, 0, 2, 0, 2, ...}
k.) an = n −
√
n2 + n
Sequences whose terms alternate signs: For alternating sequences, the
sequence converges if and only if the absolute value of the sequence goes to
zero. Moreover, if If lim |an | = 0, then lim an = 0
n→∞
n→∞
Illustration of an alternating sequence that converges to 0:
a.) an =
(
(−1)n n
n3 + 1
)∞
n=1
Illustration of an alternating sequence that diverges:
b.) an =
(
(−1)n n
3n + 1
)∞
n=1
4. Find the limit of the sequence
a.) an = {(−2)n−1 }∞
n=1
b.) an = {3 + (−0.4)n }∞
n=1
c.) an = cos(nπ) and an = sin(nπ)
d.) an =
5. Is an =
cos(nπ)
n
(−1)n + 3
an alternating sequence? What is the limit?
n3
Def: We say a sequence is bounded below if there is a number N so that
an ≥ N for all n. We say a sequence is bounded above if there is a number M
so that an ≤ M for all n. If an is bounded both above and below, then we say
the sequence is bounded.
6. Determine whether the sequence is bounded:
∞
1
a.) an =
n2 n=1
b.) an =
(
n2
n+1
∞
)∞
c.) an = {4−n }n=0
n=1
Def: We say a sequence an is increasing if an < an+1 for all n ≥ 1. We say
a sequence an is decreasing if an > an+1 for all n ≥ 1. Note: For a sequence
to be increasing or decreasing, it need not be true for ALL n, it just must be
eventually true, from some value m to ∞. If a sequence is either increasing or
decreasing, then we say the sequence is monotonic.
7. Determine whether following sequences are increasing, decreasing, or not monotonic.
a.) an =
3
n+5
b.) an = e−n
c.) an = cos
nπ
2
A recursive sequence is one in which each term of the sequence is defined
as a funtion of the previous term. To determine whether a recursive sequence
converges or diverges, find the first few terms of the sequence to see if there
appears to be a limit.
an
a.) Consider the sequence defined by a1 = 3, an+1 = 2 + . Find the first 5
3
terms of the sequence. Find the limit of the sequence.
b.) Consider the sequence defined by a1 = 2, an+1 = 1 −
terms of the sequence. Find the limit of the sequence.
1
. Find the first 5
an
12
is bounded and decreasing.
8 − an
Find the next two terms of the sequence and find the limit.
c.) The sequence defined by a1 = 4, an+1 =